Probing the ultrafast dynamics of excitons in single semiconducting carbon nanotubes

Excitonic states govern the optical spectra of low-dimensional semiconductor nanomaterials and their dynamics are key for a wide range of applications, such as in solar energy harvesting and lighting. Semiconducting single-walled carbon nanotubes emerged as particularly rich model systems for one-dimensional nanomaterials and as such have been investigated intensively in the past. The exciton decay dynamics in nanotubes has been studied mainly by transient absorption and time-resolved photoluminescence spectroscopy. Since different transitions are monitored with these two techniques, developing a comprehensive model to reconcile different data sets, however, turned out to be a challenge and remarkably, a uniform description seems to remain elusive. In this work, we investigate the exciton decay dynamics in single carbon nanotubes using transient interferometric scattering and time-resolved photoluminescence microscopy with few-exciton detection sensitivity and formulate a unified microscopic model by combining unimolecular exciton decay and ultrafast exciton-exciton annihilation on a time-scale down to 200 fs.


Supplementary note 1: Experimental setup
The experimental setup used for detecting TiSCAT and time-resolved PL signals from semiconducting (6,5) SWCNTs on glass substrates combines a pulsed laser system with a scanning confocal optical microscope (Fig. 1). A tunable Ti:Sa laser produces ≈ 150 fs pulses at a rate of 76 MHz which are split into pump and probe beams by a beam splitter (BS). For PL measurements, the probe beam is blocked. The probe pulse frequency ω probe for TiSCAT experiments is obtained by pumping a white-light generating photonic crystal fiber (PCF) followed by spectral filtering using a narrow bandpass filter (bandwidth 10 nm). The intensity of the pump pulses at the laser output ω pump is modulated by an acousto optical modulator (AOM) at a frequency of 96 kHz serving as the reference for the lock-in amplifier. The pump pulses are then sent to an optical delay stage to control the time-delay ∆t and recombined with the probe pulses using a beam splitter. The colinear beams are reflected by a beam splitter and focused onto the sample using a microscope objective with high numerical aperture (NA = 1.49) to form a tight, diffraction-limited spot. The reflected and scattered light is collected by the same objective. A longpass filter is used to suppress the pump-light. For TiSCAT, the light is guided to a sensitive photodiode (PD). The PD signal output is connected to a lock-in amplifier synchronized with the AOM signal to allow for signal demodulation at the first harmonic. In the case of time-resolved PL experiments, the light is reflected by a flip mirror (FM) towards a single-photon counting avalanche photodiode (APD) connected to time-correlated single-photon counting (TCSPC) electronics. For recording PL spectra, the light is sent to a spectrometer equipped with a sensitive charge coupled device (CCD) camera.
For the TiSCAT measurements on (6,4) SWCNTs included in this supplementary material ( Fig. 5 and 6) we utilized a tunable dual wavelength fiber laser with improved stability for excitation at 780 nm and probing at 880 nm, respectively. Unfortunately, this very stable laser system cannot be tuned to 1000 nm, the probe wavelength of (6,5) nanotubes.  The PL dynamics of 28 single (6,5) SWCNTs were investigated and their exponential exciton lifetime was determined as described in the main manuscript. Their length was determined from the corresponding confocal PL images. Whereas exciton quenching at the nanotube ends would be expected to lead to shorter lifetimes for shorter nanotubes, no clear correlation can be observed here. We explain this as follows: First, all transients were detected in the center of the nanotubes reducing the influence of end-quenching. Second, the diffraction-limited width of pump-and probe focii results in spatial averaging on a length scale of about 300 nm. This will smear out finite length effects also from short nanotubes. Third, the observed exponential lifetimes will strongly be influenced by the defect density as well as the occurence and efficiency of exciton localization in a given nanotube as mentioned above.

Supplementary note 7: Number of initially created excitons
The number of incident photons per pulse N photons can be connected to the number of initially created excitons N excitons using the absorption cross section σ of (6,5) SWCNTs. Here we take the value σ = 3.2 · 10 −17 cm 2 /C atom from ref. 6 measured at the E 22 transition (561 nm) for single (6,5) SWCNTs deposited on glass substrates with light polarized along the nanotube axis in the focus of a high NA microscope objective closely matching the experimental conditions of this study. In the present experiment, the SWCNTs are excited off-resonance at 880 nm with approx.
3 times smaller absorption efficiency as determined from the ensemble absorption spectrum in

Supplementary note 8: Monte-Carlo simulations of exciton decay
The Monte-Carlo simulation of the exciton dynamics is initialized by distributing the excitons along the one-dimensional nanotube divided into segments determined by the step size and the length of the tube (see below). The number of initially created excitons is chosen as described in sec. . The individual exciton position is assigned by a random number generation which follows a Gaussian probability distribution. The exciton distribution depicted in Fig. 9 shows the sum of 75.000 exciton position allocations and fits well with the theoretically expected Gaussian shape of the excitation focus spot geometry rendered as red line.
After the initial distribution of the excitons their stepwise movement is simulated under the initial assumption of an identical probability for a step to one or the other direction. This probability for an exciton movement is determined by the mean squared displacement and the spatial step size of the simulation. After each temporal iteration step the distance between the individual excitons along the carbon nanotube is computed and pairs of excitons whose spacing is equal or smaller than the defined exciton-exciton-annihilation distance (EEA distance) are removed from the simulation and do not interact in the further iterations. As a second decay mechanism each single exciton may decay independently after each step representing a mono-exponential decay process.
The probability for this decay type is determined by the exponential lifetime and the temporal step size of the simulation. The third decay path is given by the end quenching mechanism, which applies in the simulation in the case of an exciton reaching one end of the simulated nanotube.
After each temporal step, the number of remaining excitons is counted and stored. The whole simulation process is iteratively repeated and the temporal transient is extracted as the temporal evolution of the mean exciton number. The red data curve in Fig. 10 shows a typical transient computed from the Monte-Carlo simulation described above which fits well to eqn. 3 (black line) introduced in the main manuscript. To account for a localization site a single Gaussian-shaped energy minimum of the exciton binding energy is implemented in the simulation at the centre of the nanotube. Based on the description in ref. 9 the probability for a step to the right or the left is therefore spatially modified. The resulting decay dynamics showed a significant speed up of 13 the exciton-exciton annihilation, which can be seen in Fig. 11 as blue data curve and which is discussed in the main manuscript.